Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,18)(8,13)(9,24)(10,16)(11,17)(12,20)(14,25)(15,22)(19,33)(21,28)(23,27)(26,31)(29,35)(30,34)(32,36);(1,3,4,2)(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,19,25,34,26,23,36,35)(14,27,24,30,32,33,31,29);(2,3)(5,8)(6,13)(7,11)(9,19)(10,21)(12,15)(14,30)(16,28)(17,18)(20,22)(23,26)(24,33)(25,34)(27,31)(29,32)(35,36);(5,9,22,36,11,26,10,25)(6,14,16,31,17,32,15,24)(7,19,21,35,8,23,20,34)(12,27,13,29,28,33,18,30)Polycyclic presentation:
2^8|a^2=f,b^2=ef,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25438, #6444298, #6444301, … |
| maximal normal subgroup of | C2×#25438, #6444298, #6444301, … |
| maximal quotient of | #288109, #288114, #288118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18426);